Despite the good quality photograph of Derick, I can't make out his regiment from his capbadge. It looks like a Maltese Cross within a wreath, similar to the Rifle Brigade, but it lacks the crown so it can't be that.
Neither he nor his brother have any medal ribbons which suggests that if they did serve during WW2, neither of them did enough days to qualify for any medal.The War Medal had the least amount of qualification, which was 28 days in the armed services before 2 September 1945. The France and Germany Star only required a single day in either of those countries between 6 June 1944 and 8 May 1945.
However Derick is a Corporal which suggests that he has at least 18 months service, possibly longer. And he appears to be wearing a signaller's badge on his left cuff (crossed flags - see image below) which again suggests that he has been in the Army for long enough to qualify. Obviously based on his date of birth he could have joined up once he reached 18, at the very end of 1943, but I'm not sure exactly when he might have joined up, perhaps the Summer of 1945. Do you have a copy of Derick's wedding certificate? It's possible this will record his regiment, but probably not his Army number.
As Ross has said, David is in the RAF. Unfortunately it's not possible to identify any rank or trade badges on his uniform.
Incidentally I think there's a typo in your opening posting. David is David Herbert, hence why he was known as Bert. It was his father who was David Henry. I assume you already have the family's addresses in 1937, 1939 and 1946 (13 Latham Road, Newton Le Willows) and 1950 when their father died (149 Queen's Drive, Newton). In the 1939 register, although David's entry is open, there are two closed entries which I assume refer to Derick and Margaret (your nana).
